Question: Determine how many solutions exist for the system of equations. ${-6x-3y = -24}$ ${2x+y = -4}$
Explanation: Convert both equations to slope-intercept form: ${-6x-3y = -24}$ $-6x{+6x} - 3y = -24{+6x}$ $-3y = -24+6x$ $y = 8-2x$ ${y = -2x+8}$ ${2x+y = -4}$ $2x{-2x} + y = -4{-2x}$ $y = -4-2x$ ${y = -2x-4}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -2x+8}$ ${y = -2x-4}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.